Analysis of a one-step HMM scheme for slow-fast SDE systems
A talk in the SPDEvent series by
Jules Pertinand from Technische Universität München
| Abstract: | We aim to approximate the law of the following slow-fast stochastic differential equation (SDE) system: ${\mathrm{d}} X_t^\varepsilon = f(X_t^\varepsilon, Y_t^\varepsilon) {\mathrm{d}} t + g(X_t^\varepsilon, Y_t^\varepsilon) {\mathrm{d}} B_t \text{ in } \mathbb{R}^d$ ${\mathrm{d}} Y_t^\varepsilon = b(X_t^\varepsilon, Y_t^\varepsilon) {\mathrm{d}} t + \sigma(X_t^\varepsilon, Y_t^\varepsilon) {\mathrm{d}} W_t \text{ in } \mathbb{R}^d$ Under suitable ergodicity assumptions on the fast component $ Y^\varepsilon $, the separation of time scales leads to averaging of the slow component, i.e. $\forall t >0, \ X_t^\varepsilon \xrightarrow[\varepsilon \to 0]{\textrm{law}} \overline{X}_t$ where $\overline{X}$ satisfies the SDE ${\mathrm{d}} \overline{X}_t = \overline{f}(\overline{X}_t) {\mathrm{d}} t + \overline{g}(\overline{X}_t) {\mathrm{d}} \overline{B}_t .$ Although explicit formulas for $f$ and $g$ exist, they are costly to evaluate in practice. The classical approach is to use a so-called Heterogeneous Multiscale Method (HMM), which cleverly combines discretisation of the dynamics and approximation of $ \overline{f}$ and $\overline{g}$. Here, we propose the following one-step HMM scheme: $X_{n+1}^\tau = X_n^\tau + \Delta t f(X_n^\tau, Y_{n+1}^\tau) + \Delta t^{\frac12} g(X_n^\tau, Y^\tau_{n+1}) \Gamma_{n+1}$ $Y_{n+1}^\tau = \phi(X_n^\tau, Y_n^\tau, \tau, \gamma_{n+1})$ for $ \Delta t > 0 $, $ \tau = \sqrt{\Delta t} $, and where $ \phi $ defines a consistent ergodic scheme for $Y$. Under appropriate ergodicity assumptions for the fast dynamics and its discretisation, we show that despite the simplicity of the scheme, the approximation is not degraded compared to a standard HMM scheme: for a final time $ T > 0 $, a number of steps $ N := \frac{T}{\Delta t} $, and a test function $ \varphi $, we obtain an explicit control of the weak error $|\mathbb{E}[\varphi(X^\varepsilon_T)] - \mathbb{E}[\varphi(X^\tau_N)] | \lesssim \sqrt{\Delta t} + \varepsilon $ with a computational cost of $ \mathcal{O}(1/\Delta t) $. Joint work with Charles-Édouard Bréhier (LMAP, Pau) and Ludovic Goudenège (LaMME, Évry) |